This image below from http://www.antonine-education.co.uk is a helpful diagram to think about how displacement, velocity and acceleration are related in an oscillating system. You should be aware (in terms of displacement and period) of maximum and minimum values for all three parameters. This diagram shows how each of these parameters changes over the course of the period (2 pi radians). Note the phase shift in the plots.
Anirudh:
A great physics prof from MIT, Walter Levin, presents a great, entertaining lecture on the following topics:
Hooke's Law - Springs - Simple Harmonic Motion - Pendulum - Small Angle Approximation
As you can say, clearly relevant to this section of this topic. The video is totally accessible given what we've learned, with the exception of knowing some (basic) calculus and the following facts:
The derivative of displacement is velocity and the derivative of velocity (the double derivative of displacement) is acceleration.
Enjoy. (theres some really epic at the end too).
4.3 Forced Oscillations and Resonance
Tacoma Narrows Bridge video (please update this link)
Anirudh:
Reason for large amplitude oscillations at resonance (quoted directly from Fundamentals of Physics 7th Ed by Halliday/Resnick/Walker):
"The reason for large-amplitude oscillations at the resonance frequency is that energy is being transferred to the system under the most favorable conditions. We can better understand this by taking the first time derivative of x in Equation [x=A cos(wt)], which gives an expression for the velocity of the oscillator. We find that v is proportional to [sin(wt)]. When the applied force F is in phase with the velocity, the rate at which work is done on the oscillator by F equals the dot product F * v. Remember that “rate at which work is done” is the definition of power. Because the product F* v is a maximum when F and v are in phase, we conclude that at resonance the applied force is in phase with the velocity and that the power transferred to the oscillator is a maximum".
Comment: I included this because it requires no more than an understanding of stuff we've done so far and a basic vector property, namely the dot product. I will explain two parts of this: (1) "The rate at which the work is done on the oscillator equals the dot product F * v."
We know that this is a measure of power. Power = Energy/Time = Work done/ Time = Force * Displacement/Time = Force * Velocity.
However, Force and Velocity are vectors, so we cannot simply multiply the magnitudes. So we have to take the dot product instead!
(2) "F*v is a maximum when F and v are in phase".
Recall that the dot product of vectors A and B is IAI IBI cos X, X being the angle between the two vectors. Thus, when F and V are in phase, X = 0,so cos X =1, and thus the dot product F*v = IFI IvI, which is the maximum value for the dot product.
Answers to worksheet problems are here --> **SHM wrksht sols.tif**
This image below from http://www.antonine-education.co.uk is a helpful diagram to think about how displacement, velocity and acceleration are related in an oscillating system. You should be aware (in terms of displacement and period) of maximum and minimum values for all three parameters. This diagram shows how each of these parameters changes over the course of the period (2 pi radians). Note the phase shift in the plots.
Anirudh:
A great physics prof from MIT, Walter Levin, presents a great, entertaining lecture on the following topics:
Hooke's Law - Springs - Simple Harmonic Motion - Pendulum - Small Angle Approximation
As you can say, clearly relevant to this section of this topic. The video is totally accessible given what we've learned, with the exception of knowing some (basic) calculus and the following facts:
The derivative of displacement is velocity and the derivative of velocity (the double derivative of displacement) is acceleration.
Enjoy. (theres some really epic at the end too).
4.3 Forced Oscillations and Resonance
Tacoma Narrows Bridge video (please update this link)
Anirudh:
Reason for large amplitude oscillations at resonance (quoted directly from Fundamentals of Physics 7th Ed by Halliday/Resnick/Walker):
"The reason for large-amplitude oscillations at the resonance frequency is that energy is being transferred to the system under the most favorable conditions. We can better understand this by taking the first time derivative of x in Equation [x=A cos(wt)], which gives an expression for the velocity of the oscillator. We find that v is proportional to [sin(wt)]. When the applied force F is in phase with the velocity, the rate at which work is done on the oscillator by F equals the dot product F * v. Remember that “rate at which work is done” is the definition of power. Because the product F* v is a maximum when F and v are in phase, we conclude that at resonance the applied force is in phase with the velocity and that the power transferred to the oscillator is a maximum".
Comment: I included this because it requires no more than an understanding of stuff we've done so far and a basic vector property, namely the dot product. I will explain two parts of this:
(1) "The rate at which the work is done on the oscillator equals the dot product F * v."
We know that this is a measure of power. Power = Energy/Time = Work done/ Time = Force * Displacement/Time = Force * Velocity.
However, Force and Velocity are vectors, so we cannot simply multiply the magnitudes. So we have to take the dot product instead!
(2) "F*v is a maximum when F and v are in phase".
Recall that the dot product of vectors A and B is IAI IBI cos X, X being the angle between the two vectors. Thus, when F and V are in phase, X = 0,so cos X =1, and thus the dot product F*v = IFI IvI, which is the maximum value for the dot product.
nice extension Anirudh: Mr DJY
4.5 Wave Properties
HMWRK Assignment from Giancoli - Reflection and Refraction - Chap11 and Chap23 solutions.doc